009B Sample Midterm 2

This is a sample, and is meant to represent the material usually covered in Math 9B for the midterm. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

This problem has three parts:

(a) State the both parts of the fundamental theorem of calculus.

(b) Compute   ${\frac {d}{dx}}\int _{0}^{\cos(x)}\sin(t)~dt$.

(c) Evaluate  $\int _{0}^{\pi /4}\sec ^{2}x~dx$.

 Problem 2 

Evaluate

(a)   $\int _{1}^{2}{\bigg (}2t+{\frac {3}{t^{2}}}{\bigg )}{\bigg (}4t^{2}-{\frac {5}{t}}{\bigg )}~dt$

(b)   $\int _{0}^{2}(x^{3}+x){\sqrt {x^{4}+2x^{2}+4}}~dx$

 Problem 3 

A particle moves along a straight line with velocity given by:

$v(t)=-32t+200$

feet per second. Determine the total distance traveled by the particle

from time  $t=0$  to time  $t=10.$

 Problem 4 

Evaluate the integral:

$\int e^{-2x}\sin(2x)~dx$

 Problem 5 

Evaluate the integral:

$\int \tan ^{4}x~dx$